CAREER: Investigating Differentiated Instruction and Relationships between Rational Number Knowledge and Algebraic Reasoning in Middle School

中学数学课堂的特点是认知多样性的增加和学生学习代数的持续困难。目前，单一课堂的中学数学教学往往没有针对不同的思考者进行区分，这可能会使一些学生感到厌烦，也可能对另一些学生产生过度的挑战。学校对待同一年级不同思考者的一种常用方法是跟踪，这也被证明对学生的认知和情感都有有害的影响。此外，学生们继续努力学习代数，越来越多的中学生正在接受代数教学。本研究提出了一项新的研究和综合教育计划，通过调查(1)如何有效区分不同推理水平的中学生的教学来解决这些问题；(2)如何促进中学生代数推理与有理数知识的相互促进。该项目的教育目标是提高未来和实践教师教授认知多样性学生的能力，提高博士生对学生学习与教师实践之间关系的理解，并形成一个致力于持续专业学习如何区分教学的数学教师社区。目前正在开发三种基于研究的产品：两种学习轨迹，与教师合作开发的差异化教学材料，以及评估学生推理水平的书面评估。第一个轨迹，为学生在三个推理层次中的每一个层次阐述，重点是发展代数表达式和解决涉及有理数的基本方程；第二个学习轨迹，也为学生在三个推理层次中的每一个层次阐述，侧重于线性环境中的协变推理。此外，该项目调查了学生的课堂体验如何受到差异化教学的影响，这将允许与跟踪课堂中学生体验的研究结果进行比较。最重要的是，该项目提高了中学数学教师服务认知多样性学生的能力。项目的这一方面有可能减少机会差距。最后，该项目产生了对帮助未来教师和在职教师学习区分教学所需的各种支持的理解。该项目通过(a)将研究整合到中学数学教学中，(b)通过根据学生的认知需求定制教学来促进所有学生的学习，(c)与实践教师合作学习如何实施这种教学，(d)改进对未来数学教师和数学教育研究生的培训，在促进教学、培训和学习的同时促进发现和理解。(e)建立一个数学教师社区，他们从事持续学习以区分教学。该项目通过纳入代表性不足群体的学生，特别是有学习障碍的学生，扩大了参与范围。该项目的成果将通过会议报告广泛传播；在不同媒体发表文章；还有一个项目网站，可以提供项目产品，为媒体和公众提供项目信息，并成为促进教师与教师交流的工具。

Middle school mathematics classrooms are marked by increasing cognitive diversity and students' persistent difficulties in learning algebra. Currently middle school mathematics instruction in a single classroom is often not differentiated for different thinkers, which can bore some students or overly challenge others. One way schools often deal with different thinkers at the same grade level is by tracking, which has also been shown to have deleterious effects on students, both cognitively and affectively. In addition, students continue to struggle to learn algebra, and increasing numbers of middle school students are receiving algebra instruction. The proposed project initiates new research and an integrated education plan to address these problems by investigating (1) how to effectively differentiate instruction for middle school students at different reasoning levels; and (2) how to foster middle school students' algebraic reasoning and rational number knowledge in mutually supportive ways. Educational goals of the project are to enhance the abilities of prospective and practicing teachers to teach cognitively diverse students, to improve doctoral students' understanding of relationships between students' learning and teachers' practice, and to form a community of mathematics teachers committed to on-going professional learning about how to differentiate instruction.Three research-based products are being developed: two learning trajectories, materials for differentiating instruction developed collaboratively with teachers, and a written assessment to evaluate students' levels of reasoning. The first trajectory, elaborated for students at each of three levels of reasoning, focuses on developing algebraic expressions and solving basic equations that involve rational numbers; the second learning trajectory, also elaborated for students at each of three levels of reasoning, focuses on co-variational reasoning in linear contexts. In addition, the project investigates how students' classroom experience is influenced by differentiated instruction, which will allow for comparisons with research findings on student experiences in tracked classrooms. Above all, the project enhances middle school mathematics teachers' abilities to serve cognitively diverse students. This aspect of the project has the potential to decrease opportunity gaps. Finally, the project generates an understanding of the kinds of support needed to help prospective and practicing teachers learn to differentiate instruction.The project advances discovery and understanding while promoting teaching, training, and learning by (a) integrating research into the teaching of middle school mathematics, (b) fostering the learning of all students by tailoring instruction to their cognitive needs, (c) partnering with practicing teachers to learn how to implement this kind of instruction, (d) improving the training of prospective mathematics teachers and graduate students in mathematics education, and (e) generating a community of mathematics teachers who engage in on-going learning to differentiate instruction. The project broadens participation by including students from underrepresented groups, particularly those with learning disabilities. Results from the project will be broadly disseminated via conference presentations; articles in diverse media outlets; and a project website that will make project products available, be a location for information about the project for the press and the public, and be a tool to foster teacher-to-teacher communication.